Algebra II

Moore’s Law

The Technology Review magazine has an arresting photo essay on Moore’s Law — as told through a bunch of stunning pictures of computer chips. Click on the link above to see all the other circuits. For those who don’t know, Moore’s Law says that about every two years (some say 18 months), the number of transistors that can fit on a circuit doubles (for Wikipedia article, click here).

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The only thing I wish about the photo essay is that there was some sense of scale for each picture. Regardless, the captions tell the year each circuit was created, and the number of transistors on each circuit. The data are:

1958 1
1959 1
1961 4
1974 5000
1979 68000
1978 29000
1985 275000
1991 200000
1993 3100000
1993 2800000
2000 42000000
2007 410000000
2009 758000000

So of course, even though this data isn’t perfect nor complete, I thought I’d see how it’d look graphed.

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Ohhh, it looks like it could be exponential… Let’s plot it on a log-scale. If it’s exponential, we should get a straight line:

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Ohhh, this looks pretty linear! I wasn’t sure that it was going to work out.

The exponential line of best fit is: Transistors=e^{0.397*\text{Year}-777.29}. When I plot the data (pink) and the exponential line of best fit (blue) on the log-scale graph, you’ll see that Moore’s Law looks like it has some serious bite to it.

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Doing a little algebra with the exponential model we came up with, it appears that the number of transitors doubles about every 1.75 years.

And if you cared, Wikipedia gives their own following graphical illustration of Moore’s Law:

I’m going to be teaching exponential functions in a bit. I hope we’ll have time to do regressions. If so, I’ll probably make a 2-day investigation out of Moore’s Law.

Other posts I’ve made about logarithmic and exponential functions:

Logarithmic Graph in the News
Earthquakes, Richter Scale, and Logarithms
The Supreme Court, Linear  and Exponential Growth, and Racial Segregation
The Origin of Life on Earth and Logarithms
Paper Folding and Exponential Functions

Pendulum Lab

This week I’ve had one and a half Algebra II classes to “kill” because I’m ahead of the other teacher and we need to sync up again. Since we’re working on parabolas, I thought we could do something fun.

A while ago, I watched this video:

And I decided, perfect! I’m going do the pendulum thing in class. I got some string and washers, and put masking tape on the string every 6 inches. And I had student calculate the period of the pendulum when the length of the string was 6 inches, 12 inches, 18 inches, … , 60 inches.

To minimize the error generated by a student not exactly stopping the stopwatch when pendulum swung back and forth, I had students have the pendulum swing three times. That way any reaction time error of the person operating the stopwatch gets reduced by a third! And I had students do 3 trials for each length of string, to further minimize error.

It took all 50 minutes for students to collect all their data, plot it on a graph, and enter the data in their calculators.

Tomorrow we get to have fun. To warm up, we’re going to talk about sources of error. Then each group will get to share their graphs and talk about their findings. Then we’re going to perform a quadratic regression on our calculators, talk about if we have a good or bad model and ways to decide, and then use our model to make some predictions. (If we know the period, can we find the length of the pendulum, and vice versa.) Then, I’m going to conclude by showing students the theoretical formula for the period of a pendulum (T=2\pi\sqrt{\frac{L}{g}}) and we’re going to see if their collected data matches up with the theoretical predictions.

The best part about this is that all the groups data seems in line with each other, and in fact, they are all really close to the theoretical predictions. I can’t wait to see if there are oohs and aahs about how accurate these data points they got are to the theoretical predictions.

I’m excited for tomorrow!

UPDATE: my data collection sheet (PDF), my lab debrief sheet (PDF).

Inequalities and Quadratics

In Algebra II,, we’ve recently been delving into quadratics. I recently blogged about how I taught completing the square and the quadratic formula, and put up a bunch of resources. Since then, we’ve moved on to graphing quadratics, followed by inequalities.

The complete topic list for inequalities is:

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I’ve been trying something new, which is creating packets for students to work on. In essence, I’m creating my own textbook for these sorts of questions. I thought I’d share them with you in case they prove useful [1]. I’m pretty proud of them — and they way they fit together and build up understanding, not just providing a method to solving problems.

1. PACKET 1: Linear and Quadratic Inequalities on the Number Line (PDF version)
2. Additional Homework on Quadratic Inequalities (PDF version)
3. PACKET II: Linear and Quadratic Inequalities on the Coordinate Plane (PDF version)

4. PACKET III: Systems of Inequalities (Linear and Linear-Quadratic) (PDF version)
5. Additional Homework on Systems of Inequalities (PDF version)
6. Pop Quiz on Inequalities and Quadratics (PDF version)

Hopefully they’ll be useful to someone else out there!

[1] The formatting might be a bit off for you… It looks slightly off (meaning the pages don’t end where I intended them to end) on my mac but fine on my PC. I think you need to make sure that on a Mac you select all and convert the font to “Gill Sans” (on a PC, I think it’s called “Gill Sans MT”, which is creating the problem).

UPDATE: PDFs posted, without typographic weirdnesses.

Completing the Square

Yesterday I ahem-ed and winked to my Algebra II class about them needing to know how to complete the square for class today. Teaching this topic last year was a nightmare. A total trainwreck. Students were having difficulty all over the place — they couldn’t simplify radicals, they didn’t get why the procedure worked, they were wondering how imaginary numbers came into play here, they confused the steps, they didn’t *get* it. And it was my fault.

Part of the problem was that we were doing too much, too fast. We had brought in graphing quadratics early on, and we were emphasizing the relationship between the equations and the graphs from the start. We also — in the middle of the quadratic unit — taught complex numbers. That’s too many huge things to deal with. Quadratics bring too much together, and we needed to keep the ideas and skills organized so they make sense.

So the other Algebra II teacher and I decided we’d try something different. First, this year, we introduced complex numbers without talking about quadratics. We motivated these numbers, and then we had students practice working with them, getting really comfortable with them. Second, when we started quadratics, we did so without any graphing. Period. We were doing all algebraic work.

Here’s how we progressed.

PART I: Review
Regular, very simple equations with solutions involving square roots, imaginary numbers, and real numbers:

quad1
Quick review of factoring:

quad2

A brief discussion of solving equations with perfect square terms — with imaginary and real solutions:

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Part II: Completing the Square

Perfect Squares:

quad4

We talked about what a perfect square is and noticed a relationship between the four terms — when you FOIL. Importantly, students are going to see that the second and third terms are the same.

***

Creating Perfect Squares!:

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The next step of creating perfect squares really has them grapple with the fact that the missing constant term is simply half of the coefficient of the x term squared.

***

Completing the Square:

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For me, it was this step, just a short distance from the last step, which made the entire unit a success. Because now my students had seen the relationship between all the terms in a perfect square and actually seemed to understand them. My favorite part was that most students were getting problem 10 right — and it involves fractions! We also talked about how important signs are for this process.

***

The End Game: Completing the Square

Before we actually “completed the square” I had students look at the last section of the review sheet.

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We talked about how if you can write a problem in this form, that you can ALWAYS solve it. And what we were going to be doing is finding a way to write any quadratic equation in that form, so we can solve it.

Then, I went through an example — step by step — to get a problem to that form:

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Then I had them solve it, like they had done previously. Most of them had no trouble solving it.

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They practiced doing a few problems on their own — some which gave “nice” answers, some which gave answers with radicals, and some which gave complex answers.

Part III: Reinforcement

I made them practice a few more times, with some harder problems, and then I threw them a curve ball — a coefficient in front of the x^2 term. We conquered that, although there were same difficulties with fractions. Then I put some terms on one side and some terms on the other side (e.g. x^2=2x-15).

Overall, they really rocked it. How do I know they got it?

I started off this post by saying that I ahem-ed about giving a pop quiz in my class today. Well, I followed through on that. gave a pop quiz to my class on completing the square. I gave them one easy problem and one much more difficult problem — with fractions and radicals — on completing the square.

I got a whole bunch of perfect scores.

If you want, the worksheets I created are below:

Factoring Quadratics
Completing the Square, Part I
Completing the Square, Part II
Completing the Square, Pop Quiz

Linear Regressions

In Algebra II tomorrow, I’m going to finish up talking about linear regressions. And in honor of that, I’ve created their Thanksgiving homework: a worksheet.

I’m having students

1. pick 10 words
2. check to see how many wikipedia hits these words got in October 2008
3. check to see how many google hits there are for those words

(Example: “monkey” was looked up 122,694 in October 2008 and has about 140,000,000 google hits.)

Then they’re going to see if the data looks linear, and calculate a line of best fit, and use it to predict.

Even though the worksheet needs a lot of work in terms of the questioning and phrasing (I am so tired that I couldn’t think of great questions… I just felt the need to pound this out), I still think this is one of my better activities.

If you want: 2008-11-25-worksheet-on-linear-regressions

Perhaps I’ll compile all the data from all the students and we’ll have a larger data set. We’ll get to talk about outliers (e.g. if you look the word “water” up, things are crazy). I personally am curious what will happen.

(FYI, for the 10 words I chose, I got an r value of about 0.7.)

Advice for using an online math textbook

In this Year Of Massive Transformations in my school (many new faculty, new administrative structure with loads new administrators, a new department head for me), we’re also overhauling the high school math curriculum. We’re really trying to come up with a great Algebra II/Precalculus sequence, and I’m involved with helping codify the non-accelerated track. We’re definitely switching textbooks (the one we’re using now is just too hard for the kids).

In our search, we came across an Algebra II textbook published by Holt. We liked the examples, the number and kinds of homework problems, the layout, and the sequencing. (Although we’ll deviate from the sequencing a bit.) The best part about it: if we buy the textbook (around $80), we get access to the e-book for 6 years for free. And from what I understood from talking to the representative, we can just pass on the password from student to student from year to year.

Our Student Council is soon going to be approaching department heads about getting e-books for some of the courses (the physical books are really heavy and expensive). It makes really good sense because we’re a laptop school! I’m going to request that the school purchase these books and charge students $20/year for access to the e-book. And then for students who want to borrow a physical textbook, they can get them from us.

But this all seems very logistically challenging. I can anticipate a few problems already (importantly: what do you do with the excuse “I didn’t have internet access where I was last night”?)

Which is why I bring this to you. Have any of you used online textbooks before? Anything I should keep in mind when making this decision? Any great benefits to it? Any great drawbacks?

And if you haven’t used online textbooks, what sort of problems would you anticipate?

Mathematics Illuminated & the Carnival of Education

1. The Carnival of Mathematics 43 is out. There’s some really great stuff there! Including a really wonderful problem for an Algebra II class! And a great way to do test review!

2. Today in my MV Calculus course, I was teaching curvature. One of my students asked for the dimensions of curvature. Love those sorts of questions! In any case, when I was looking online for some good resources, I came across this website which explains curvature — and a bunch of other really interesting math topics — to the layperson.

So, here’s my present to you: if you’re a math teacher and you have an extra class to introduce the ideas behind advanced mathematics, without going into all the equations and nuances, you have your lesson plan laid out here, at Mathematics Illuminated. Totally awesome stuff! Plus, if you register (for free!), you can stream videos on teach topic. Unfortunately, I haven’t been able to watch one yet, so tell me if you get a chance if the videos are any good in the comments.